Analytical approximation for solidification processes in PCM storage with internal fins: imposed heat flux

Uses of thermal energy storage systems have expanded notably in recent decades. In thermal energy systems, internal heat transfer enhancement techniques such as fins are often used because of the low thermal conductivity of the phase change materials (PCMs). In this paper, solidification of a PCM is studied in a rectangular storage with horizontal internal plate fins and an imposed constant heat flux on the vertical walls. A simplified analytical solution is presented and its results are compared to those for a numerical approach based on an enthalpy method. The fraction of solidified PCM in storage is calculated with the derived analytical model which determines how much of the storage is solidified after a certain time. The results show that the analytical model satisfactorily estimates the solid–liquid interface and the temperature distribution for the fin, which are useful in the design of PCM-based thermal energy storage or cooling systems.     

Analytical model for melting in a semi-infinite PCM storage with an internal fin

The most PCMs with high energy storage density have an unacceptably low heat conductivity and hence internal heat transfer enhancement techniques such as fins or other metal structures are required in latent heat thermal storage (LHTS) applications. Previous work has concentrated on numerical and experimental examination in determining the influence of the fins in melting phase change material. This paper presents a simplified analytical model based on a quasi-linear, transient, thin-fin equation which predicts the solid-liquid interface location and temperature distribution of the fin in the melting process with a constant imposed end-wall temperature. The analytical results are compared to the numerical results and they show good agreement. Due to the assumptions made in the model, the speed of the solid-liquid interface during the melting process is slightly too slow.     

Modeling the rubbing contact in honeycomb seals

Metallic honeycomb labyrinth seals are commonly used as sealing systems in gas turbine engines. Because of their capability to withstand high thermo-mechanical loads and oxidation, polycrystalline nickel-based superalloys, such as Hastelloy X and Haynes 214, are used as sealing material. In addition, these materials must exhibit a tolerance against rubbing between the rotating part and the stationary seal component. The tolerance of the sealing material against rubbing preserves the integrity of the rotating part. In this article, the rubbing behavior at the rotor–stator interface is considered numerically. A simulation model is incorporated into the commercial finite element code ABAQUS/explicit and is utilized to simulate a simplified rubbing process. A user-defined interaction routine between the contact surfaces accounts for the thermal and mechanical interfacial behavior. Furthermore, an elasto-plastic constitutive material law captures the extreme temperature conditions and the damage behavior of the alloys. To validate the model, representative quantities of the rubbing process are determined and compared with experimental data from the literature. The simulation results correctly reproduce the observations made on a test rig with a reference stainless steel material (AISI 304). A parametric study using the nickel-based superalloys reveals a clear dependency of the rubbing behavior on the sliding and incursion velocity. Compared to each other, the two superalloys studied exhibit a different rubbing behavior.     

An engineering procedure for air side performance evaluation of flat tube heat exchangers with louvered fins

An accurate evaluation of possible air side heat transfer surface geometries is a prerequisite for optimal heat exchanger design. Aiming for practical engineering applicability a simplified and transparent analytical procedure for the assessment of louvered fin and flat tube heat exchanger geometries and the calculation of fin parameters that enable maximal performance for given boundary conditions has been developed. The proposed method comprises determining fins temperature profiles and effective heat transfer temperature difference, introduction of a relative heat transfer surface area, as well as the utilization of recent experimentally obtained heat transfer correlations confirmed for the observed range of boundary conditions. The proposed methodology is validated through comparison with experimental and numerical results of other authors.     

Optimization of convective fin systems: a holistic approach

 A numerical analysis of natural convection heat transfer and entropy generation from an array of vertical fins, standing on a horizontal duct, with turbulent fluid flow inside, has been carried out. The analysis takes into account the variation of base temperature along the duct, traditionally ignored by most studies on such problems. One-dimensional fin equation is solved using a second order finite difference scheme for each of the fins in the system and this, in conjunction with the use of turbulent flow correlations for duct, is used to obtain the temperature distribution along the duct. The influence of the geometric and thermal parameters, which are normally employed in the design of a thermal system, has been studied. Correlations are developed for (i) the total heat transfer rate per unit mass of the fin system (ii) total entropy generation rate and (iii) fin height, as a function of the geometric parameters of the fin system. Optimal dimensions of the fin system for (i) maximum heat transfer rate per unit mass and (ii) minimum total entropy generation rate are obtained using Genetic Algorithm. As expected, these optima do not match. An approach to a `holistic' design that takes into account both these criteria has also been presented. Received on 22 February 2001 / Published online: 29 November 2001     

Optimization of annular fins on a horizontal tube

A combination of uniform-thickness annular fins evenly spaced on a tube is a common extended-surface heat exchanger configuration. An analytical model is developed and is verified by comparing total heat transfer predicted by the model to available experimental data. A direct-pattern search technique is applied to the model to optimize the fin/ tube geometry. Optimum dimensions and spacing of fins are established to provide the maximum free convection heat transfer from a fin/tube combination. The optimum arrangement is dependent on fin thermal conductivity, tube diameter, volume of fin material per unit length of tube, and temperature difference between the tube and the surrounding air. Calculated results indicate that a fin in the optimum fin/tube system is shorter and thicker than an isolated fin optimized for minimum material (with no consideration of the effects of fin spacing).     

Viscoelastic analysis of rectangular passage of microchanneled plates subjected to internal pressure

The micro heat exchangers or micro reactors in miniature chemical and thermal systems entail the design know-how of microchanneled plates or miniaturized multichannel passages with fins. With the increase of temperature and pressure in the chemical and thermal systems, the design of the miniature components against creep failure becomes a primary concern. To develop a simplified design method, the time-dependent deformation and stresses of the plate–fin structures are analyzed by assuming the fins as elastic springs. Using the numerical inversion of Laplace transform and the standard linear solid model, the analytic expressions of the deformation and stresses are derived. The feasibility and accuracy of the method are verified by a comparison of the analytical results with finite element simulations.     

Thermal analysis of annular fins with temperature-dependent thermal properties

The thermal analysis of the annular rectangular profile fins with variable thermal properties is investigated by using the homotopy analysis method （HAM）. The thermal conductivity and heat transfer coefficient are assumed to vary with a linear and power-law function of temperature, respectively. The effects of the thermal-geometric fin parameter and the thermal conductivity parameter variations on the temperature distribution and fin efficiency are investigated for different heat transfer modes. Results from the HAM are compared with numerical results of the finite difference method （FDM）. It can be seen that the variation of dimensionless parameters has a significant effect on the temperature distribution and fin efficiency.     

Three-dimensional numerical investigation of heat transfer for plate fin heat exchangers

In the present study, the potential of rectangular fins with 30° and 90° angle and 10 mm offset from the horizontal direction for heat transfer enhancement in a plate fin heat exchanger is numerically evaluated with conjugated heat transfer approach. The rectangular fins are mounted on the flat plate channel. The numerical computations are performed by solving a steady, three-dimensional Navier–Stokes equation and an energy equation by using Fluent software program. Air is taken as working fluid. The study is carried out at Re = 400 and inlet temperatures, velocities of cold and hot air are fixed as 300, 600 K and 1.338, 0.69 m/s, respectively. Colburn factor j versus Re design data is presented by using Fluent. The results show that the heat transfer is increased by 10 % at the exit of channel with fin angle of 30° when compared to channel without fin for counter flow. The heat transfer enhancement with fins of 30° and 90° for different values of Reynolds number with 300, 500 and 800 and for varying fin heights, fin intervals and also temperature distributions of fluids on the top and bottom surface of the channel are investigated for parallel and counter flow.     

Analysis of heat transfer in a partially wet radial fin assembly during dehumidification

This paper presents the analysis of heat transfer in a partially wet annular fin assembly during the process of dehumidification. In past studies, both fully dry and fully wet fins have been analyzed. New analytical formulation leading to a closed-form solution has been developed for a partially wet fin, which is most common in dehumidifier coil operation during air conditioning. The parameters that influenced the heat transfer rate in the finned tube structure are ratio of fin and wall thermal conductivities, ratio of fin thickness to fin pitch, ratio of wall thickness to fin pitch, ratio of fin length to fin pitch, cold fluid Biot number, ambient Biot number, the relative humidity and dry bulb temperature of the incoming air, and the cold fluid temperature inside the coil. Calculations were carried out to study the performance of the heat exchanger. The computed results included the temperature distribution in the wall and the fin and the fin efficiency.     

Experimental study of convective heat transfer from a rotating finned tube in transverse air flow

 The convective heat transfer from fins to air has been evaluated using rotating annular fins subjected to an air flow parallel to the fins. The fin cooling is studied using infrared thermography. The thermal balance in a fin during its cooling process allows us to obtain the heat transfer coefficient from the temperature time evolution of the fin. Moreover, Particle Image Velocimetry allows us to obtain the flow field in the mid-plane between two fins. The influence of the fin spacing on the convective heat transfer is studied for various velocities of the superposed air flow and various fin rotational speeds. These tests were carried out for air flow Reynolds numbers (based on the shaft diameter and the velocity of the superposed air flow) between 2550 and 18200 and rotational Reynolds numbers (based on the shaft diameter and the peripheral speed) between 800 and 2.9 × 10⁴, for different fin spacings. Received: 14 May 1999/Accepted: 8 October 1999     

Natural convection heat transfer from a thin rectangular fin with a line source at the base — a finite difference solution

This paper presents an analysis of the problem of a thin fin of finite thermal conductivity, with an isothermal line source at the base, dissipating heat to the surrounding air by natural convection. The horizontal surface to which the fin is attached is adiabatic so that heat is dissipated only through the fin. The temperature and velocity distributions in the field, the temperature profile in the fin, local Nusselt numbers along the fin and the average heat transfer coefficient of the fin are obtained by solving the governing equations in the field and the heat transfer equation in the fin simultaneously, using an explicit unsteady Finite Difference formulation leading to the steady state result. Numerical experiments are performed to study the influence of parameters namely the fin height, temperature of the heating source and the fin material on the average heat transfer coefficient. Comparison is made with fins of infinite thermal conductivity and the vertical isothermal flat plate.     

Inverse determination of the heat transfer characteristics on a circular plane fin in a finned-tube bundle

In this work, we present the numerical results of the average heat transfer coefficients, $ \overline{{h_{\varphi } }} $ , over a circular plane fin in a finned-tube bundle for both aligned and staggered arrangements as well as the fin efficiency and the heat flux dissipated from the whole fin. The study covers a wide range of Reynolds number (2 × 10³–3 × 10⁴), for three different positions of the finned tube inside the heat exchanger. The temperature distribution on the fins surfaces was obtained experimentally using infrared thermography technique. The predicted values of the heat transfer coefficient were obtained numerically using the finite element method in conjunction with the conjugate gradient algorithm and the measured temperatures.     

A numerical study of the thermal performance of fins with time-dependent boundary conditions,including initial transient effects

A finite-volume numerical model is constructed and applied to evaluate the time-dependent heat-transfer characteristics of longitudinal fins with a rectangular profile. To facilitate comparisons with analytical solutions, the problem is taken to be one-dimensional with constant thermal properties. Two base boundary conditions are considered: (1) a step change and (2) a periodic change in temperature. Parametric studies are carried out to investigate the effects of grid size, time step, fin parameter, Fourier number, amplitude and frequency of base-temperature oscillation on the fin temperature distribution, heat-transfer rate and efficiency. The study shows an excellent agreement between the numerical and analytical results and investigates the importance of the initial transient effects.     

Optimization of heat transfer through finned dissipators cooled by laminar flow

In the present work, the problem of optimizing the shape and the spacing of the fins of a thermal dissipator cooled by a fluid in laminar flow is studied. For a particular finned conduit, the velocity and temperature distributions on the transversal section are determined with the help of a finite element model and a global heat transfer coefficient is calculated. A polynomial lateral profile is proposed for the fins and the geometry is optimized in order to make the heat transfer coefficient as high as possible with the smallest dimensions or the lowest hydraulic resistance to the flow. The optimum fin profile and spacing, obtained by means of a genetic algorithm, are finally shown for different situations. Increases of 45% are obtained in the heat transfer coefficient referring to the maximum values which can be obtained with rectangular fin profiles.     

Analytical and numerical solution along with PC spreadsheets modeling for a composite fin

Heat transfer through composite fins is investigated by both analytical and numerical methods. In this regard, governing differential equations of the two dimensional fin and one dimensional cladding are studied to examine the effect of Biot number and ratio of thermal conductivities of the fin material to the cladding, on the dimensionless temperature profiles. The results show that one dimensional analysis, traditionally used in fin analysis, is not applicable for composite fins, particularly when the conductivity ratio of the composite fin materials is low. In addition, the use of spreadsheet programs in solving the fin problem is investigated in somewhat more detail with regard to the solution as well as presentation of the graphical results.     

Correlation equations for optimum design of annular fins with temperature dependent thermal conductivity

Correlation equations for optimum design of annular fins with temperature-dependent thermal conductivity are obtained in the present work. The nonlinear fin equation which is associated with variable thermal conductivity condition is solved by Adomian decomposition method that provides an analytical solution in the form of an infinite power series. The optimum radii ratio of an annular fin which maximizes the heat transfer rate has been found as a function of Biot number and the fin volume for a given thermal conductivity parameter describing the variation of the thermal conductivity. The fin volume is fixed to obtain the dimensionless geometrical parameters of the fin with maximum heat transfer rate. The data from the present solutions is correlated for a suitable range of Biot number and the fin volume. The simple correlation equations presented in this work can assist for thermal design engineers for optimum design of annular fins with temperature-dependent thermal conductivity.     

Heat transfer from extended surfaces subject to variable heat transfer coefficient

The present article investigates the effect of locally variable heat transfer coefficient on the performance of extended surfaces (fins) subject to natural convection. Fins of different profiles have been investigated. The fin profiles presently considered are namely; straight and pin fin with rectangular (constant diameter), convex parabolic, triangular (conical) and concave parabolic profiles and radial fins with constant profile with different radius ratios. The local heat transfer coefficient was considered as function of the local temperature and has been obtained using the available correlations of natural convection for each pertinent extended surface considered. The performance of the fin has been expressed in terms of the fin efficiency. Comparisons between the present results for all fins considered and the results obtained for the corresponding fins subject to constant heat transfer coefficient along the fin are presented. Comparisons, i.e. showed an excellent agreement with the experimental results available in the literature. Results show that there is a considerable deviation between the fin efficiency calculated based on constant heat transfer coefficient and that calculated based on variable heat transfer coefficient and this deviation increases with the dimensionless parameter m.     

Numerical analysis of laminar natural convection in enclosures with fins attached to an active wall

Vertical enclosures with conducting fins attached to the cold wall were considered. Side walls were kept at constant but different temperatures, while horizontal top and bottom walls were insulated. A conjugate formulation was used for the mathematical formulation of the problem, and a computer program based on the control volume approach and the SIMPLE algorithm was developed. Computations were performed to investigate the effects of the fin configuration and Rayleigh number on the flow structure and heat transfer. It was observed that the heat transfer rate through an enclosure can be controlled by attaching fins to the wall(s) of the enclosure. At low Rayleigh numbers (conduction controlled regime), the heat transfer rate increases with the increasing number of fins and the fin length. However, at higher Rayleigh numbers (convection dominant regimes), the heat transfer rate can be decreased or increased by properly choosing the number of fins and the fin lengths. Received on 07 April 1997     

An experimental investigation on performance of rectangular fins on a horizontal base in free convection heat transfer

This paper reports an experimental study of free convection heat transfer from rectangular fin-arrays on a horizontal base. An experimental set-up was constructed and calibrated, 15 sets of fin-arrays and a base plate without fins were tested in atmosphere. Fin height was varied from 6 mm to 26 mm, fin spacing was varied from 6.2 mm to 83 mm. The base-to-ambient temperature difference was also varied independently and systematically with the power supply to heater ranging from 8 W to 50 W. Fin length and fin thicknesses were fixed at 100 mm and 3 mm, respectively. The experimental program was conducted so as to clearly delineate the separate roles of fin height, fin spacing and base-to-ambient temperature difference. It was found that for a given base-to-ambient temperature difference the convection heat transfer rate from fin-arrays takes on a maximum value as a function of fin spacing and fin height. For a given base-to-ambient temperature difference the enhancement of the convection heat transfer rate of fin-arrays relative to that for base plate without fins is strongly dependent on the fin spacing to fin height ratio and number of fins. A correlation was also presented relating the convection heat transfer rate of fin-arrays relative to that for base plate without fins with the relevant non-dimensional parameters. Received on 7 August 1997